In big Rudin's book, it constructs the Lebesgue measure by first defining a positive functional, and then using Riesz representation theorem. It arises me to think that if every measure can be constructed by positive functional on locally compact Hausdorff space, if not, what kind of measure can be constructed by such method?
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1Keep reading, and you will find answers in chapter 6! – Ningxin Feb 11 '16 at 18:00
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What I should add is that given a measure $\mu$, you can of course define a linear functional $\Lambda f = \int fd\mu$. – Ningxin Feb 11 '16 at 18:13
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@QiyuWen: That's true. What's more interesting, and maybe what the OP wants to know, is that in a general LCH space, there can exist distinct measures $\mu,\nu$ which define the same positive linear functional $\Lambda$, i.e. $\int f,d\mu = \int f,d\nu$ for all continuous compactly supported real-valued $f$. Of course, by the Riesz representation, for a given $\Lambda$ there is exactly one corresponding measure which is regular, but there can be many more which are not regular. Rudin doesn't really discuss this, as far as I can tell. – Nate Eldredge Feb 11 '16 at 19:18
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@NateEldredge any example if non-regular? – 89085731 Feb 11 '16 at 19:45
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The standard example is the Dieudonné measure on the uncountable ordinal space $[0 ,\omega_1]$. See http://math.stackexchange.com/a/161396/822 for a reference. You won't find a counterexample on a "nice" space - on any locally compact metric space, a locally finite Borel measure is automatically regular. – Nate Eldredge Feb 11 '16 at 23:08